On some problems, it makes sense to try some reasonable guesses and see what happens.

Example 1

Problem: Find all real roots of the polynomial

\[x^3-3x^2+49x-147.\]

To do this, we can remember the following helpful fact: if \(k\) is an integer root of a polynomial, then \(k\) must divide the degree zero coefficient. Thus what we are looking for is one of the divisors of \(147\). The divisors of \(147\) are \(\pm1, \pm3, \pm7, \pm21, \pm49,\) and \(\pm147\)

For simplicity, we start by guessing the smaller ones. It turns out \(\pm 1\) are not roots, but \(3\) is a root. Therefore by long division we can factor

\[x^3-3x^2+49x-147 = (x-3)(x^2+49).\]

From this we see that the only real root of the polynomial is \(3\).